Find the most general form of the antiderivative of f(x)= xe^(x^2).

To solve the integral `int xe^{x^2}dx` consider the substitution `u=x^2`. Then `du=2xdx` which means the integral becomes

`1/2 int e^udu`

`=1/2 e^u + C`

`=1/2 e^x^2 +C`

If you have two functions multiplied together like `x` and `e^{x^2}` , where the derivative of the argument of the one function `(d/{dx}(x^2)=2x)` is the other function, then we can always make the substitution which makes the integral easier to solve.

To solve the integral `int xe^{x^2}dx` consider the substitution `u=x^2`. Then `du=2xdx` which means the integral becomes

`1/2 int e^udu`

`=1/2 e^u + C`

`=1/2 e^x^2 +C`

If you have two functions multiplied together like `x` and `e^{x^2}` , where the derivative of the argument of the one function `(d/{dx}(x^2)=2x)` is the other function, then we can always make the substitution which makes the integral easier to solve.